The reason the CC doesn't continue to climb is that more chlorine oxidizes the CC creating end products that are no longer CC. In the case of ammonia, chlorine reacts quickly with it to form monochloramine which is CC, but then it further reacts with it more slowly to form dichloramine (still a CC) which reacts with the monochloramine to form nitrogen gas and hydrochloric acid neither of which are CC. So there is no longer any CC since the chlorine is no longer bound to the nitrogen and instead got reduced to chloride salt.
And as I wrote and you understood, some CC is broken down by sunlight either directly or indirectly and here again the chlorine that was bound to ammonia or the organic gets converted (eventually) to chloride salt. So the "buildup" from chlorine usage/consumption is chloride salt (technically sodium chloride salt because of the sodium from chlorinating liquid or bleach that balances the chlorine in hypochlorite). The oxygen from the hypochlorite ends up as either water (for oxidation of ammonia and organics) or as oxygen gas (when chlorine breaks down from sunlight). Technical details are in
this post.
And yes, when you see CC buildup in your pool it most likely means that you are introducing ammonia and organics (ones that can combine with chlorine) faster than the sum of sunlight breaking it down or chlorine further oxidizing it is able to keep up. Assuming you are maintaining proper chlorine levels, the easiest way to fix this is to expose the pool to sunlight longer. For an indoor pool or any pool not exposed to sunlight one can add a UV system.
Urine is indeed particularly bad because it contains a lot of nitrogenous components that can react with chlorine, especially urea, ammonia, and creatinine, but also amino acids (see
this post for more details). One cup of urine is equivalent to 5 person-hours of bather load. Urine is also worse than sweat because it contains a higher content of urea and urea reacts rather slowly with chlorine so builds up unless sunlight is present (we think because sunlight breaking down chlorine creates hydroxyl radicals that oxidize the urea before it gets a chance to react with chlorine).
Superchlorination (i.e. raising the FC/CYA ratio) will help get rid of CC for CC that is ammonia based (i.e. inorganic chloramines), but can make the problem worse creating more CCs for slow-to-oxidize organics such as urea especially if it has built up in the pool. If the introduction of urea is stopped, then superchlorination can eventually get rid of the urea and the CCs but this can be slow. Some commercial/public pools try and do this by superchlorinating at night but this is not always successful due to the high bather-loads during the day.
There is no such thing as breakpoint for a pool because you are always maintaining chlorine in the pool. The term "breakpoint" comes from adding chlorine to ammonia in terms of how much chlorine is needed to get over the hump to oxidize it. However, that is not the situation in the pool because you are always adding chlorine to maintain its level and the much smaller amount of ammonia is only slowly getting introduced so "breakpoint" is continuous. The concentration of active chlorine (proportional to the FC/CYA ratio) determines how quickly the chemical reactions with chlorine occur, but there isn't some magic concentration where all of a sudden things start to happen. It's continuous where higher chlorine levels make the reactions go faster. The reaction rate is determined by the concentration (so proportional to the FC/CYA ratio) but the stoichiometric quantity of chlorine that is required gets taken from the chlorine reserve (i.e. the FC which is mostly chlorine bound to CYA) and is replenished when you add more chlorine.
However, there is a downside for using chlorine to oxidize bather waste because for some of the reactions higher chlorine levels can produce more nitrogen trichloride that is the most volatile and irritating chloramine. In fact, there is a substantial amount of nitrogen trichloride produced from the chlorine oxidation of urea which is why sunlight is important to get rid of the urea rather than have chlorine try and oxidize it. See
this post for more technical details.